The phrase “at the other end” implies the context that there is something with two recognisable ends, such as a corridor, a street, a candle or a length of rope. It also implies that the focus of attention is at one end - for example, both speaker and listener might be at the South end of a corridor that runs North-South.
“The other end” then refers to the end not previously focused on. So if the focus (whether express or implied) was on the South end, the other end would be the North end. By implied focus, I’m referring to situations such as the following: a fireman breaks down the front door to an apartment at the South end of a corridor, and shouts to the person inside that here is a fire at “the other end” of the corridor. Although the fireman didn’t actually say, “We are at this end” or “We are at one end”, the physical and very obvious location of both speaker and listener would be understood to be “this end”.
Example: one end of a candle has an exposed wick. The other end is flat.